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Traditional prohibitions are set in the 2 x 2 grid, i.e. don't allow Attribute 1 level 1 to show up with Attribute 2 level 1. Alternative-specific designs are where we knock out an entire row or column, i.e. don't allow any level of Attribute 2 to show with Attribute 1 level 1.

So, the initial way I thought to do this would be to collapse A and B down into a single attribute with 4 levels:
off/off
on/off
off/on
on/on

Then you could enter in the prohibitions as usual and prohibit all levels of C from showing up with levels 1-3 of the new, bigger attribute A/B.

Then I re-read the post where you say you set the prohibitions, so I gave it a whirl in the Advanced Prohibitions tab to specify a complex within-concept prohibition (I've never tried it there before). There is a warning that comes up that says "Although this is allowed, such prohibitions may make it difficult for the designer to find proper alternative-specific designs." But, upon clicking to proceed, I indeed get a message that the task could not be generated according to the constraints.

So, the likely explanation is that the designer algorithm has some issues with interpreting these types of prohibitions, and it's more of an internal code issue that really a tough design issue. So, I'll step back and go with my initial suggestion of collapsing down the attributes.

Combining A and B was was my first thought but unfortunately they both have 4 levels (including 1 off state) i.e.

A: Off, On1, On2, On3
B: Off, On1, On2, On3

Thefore collapsing into one attribute would result in 16 levels.

For the time being I have got two C attributes (CA and CB) which relate to attribute A and B respectively and made them alternative specific. In the context of the quesiton this makes sense but just adds more granularity than is needed.

If it is a designer algorithm issue, could you manually build a design instead? Sounds a bit dangerous so we probably wouldn't do that, but just out of interest.

SSI Web does allow you to import designs from a CSV file, but the tough part would be coming up with the design to begin with. You can always run the design tests after you import something.

Another approach might be to go ahead and generate the design using the 16 levels and then export it to a CSV file. You could then recode it back into a 2 attribute design, i.e. level 1 in the big design would be Off/Off, level 2 would be off/one1, and so forth. Then you would recode that one column into two columns where a 1 becomes a 1,1 and a 2 becomes a 1,2 and so forth.

You can make attribute C be conditional on A and B without using an alternative specific design. Within SSI Web there is a feature called "Conditional Relationships." You can access this from the Attributes tab of the CBC Exercise dialog.

Is this not just for conditional displays and conditional pricing? I want to make the presence of an attribute conditional on two primary attributes, rather then just conditionally displaying text or a price.